international journal of heat and mass...

International Journal of Heat and Mass Transfer 54 (2011) 4913–4922

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International Journal of Heat and Mass Transfer

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Evaporation characteristics of fuel droplets with the addition of nanoparticlesunder natural and forced convections

Yanan Gan, Li Qiao ⇑School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 May 2011Received in revised form 28 June 2011Accepted 28 June 2011Available online 23 July 2011

Keywords:NanofluidsNanoparticleFuel dropletEvaporation rateD2-lawParticle aggregation

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.07.003

⇑ Corresponding author. Address: 701 W. Stadiu47907-2045, USA. Tel.: +1 765 494 2040; fax: +1 765

E-mail address: [email protected] (L. Qiao).

Evaporation characteristics of fuel droplets containing suspended aluminum nanoparticles were deter-mined under natural and weak forced convections at varying flow temperatures. The results show thatunder higher convection temperatures droplet evaporation follows the classical D2-law. Under low forcedconvection temperatures or natural convection, however, a departure from the classical D2-law wasobserved and the evaporation rate gradually slowed. A modeling based on the population balance equa-tion was performed to understand the dynamic particle aggregation process. The modeling results sug-gest that particle aggregation is likely the cause of the D2-law-deviation phenomena.

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1. Introduction particles in a shock tube, demonstrating that the addition of

Nanofluids, an exciting new class of nanotechnology-based heattransfer fluids, have received much interest in recent years [1–3].They are liquids with the stable suspension of a small amount ofnanometer-sized particles (e.g., metals, oxides, carbides, nitrides,or carbon nanotubes) having typical lengths of 1–100 nm. Nanofl-uids have been discovered to have significantly enhanced thermalconductivity and can be used for a variety of energy/thermal sys-tems, such as electronic cooling and microelectromechanical sys-tems (MEMS). Besides heating and cooling purposes, nanofluidscan also be applied in other areas. Senthilraja et al. [4] provideda comprehensive review of the applications of nanofluids in auto-mobiles as a coolant, fuel additive, lubricant, shock absorber, andrefrigerant.

The present study is based on a new concept in the field of com-bustion and fuels: tailored nanofluid-type fuels created by properlymixing nanoscale materials such as energetic metals with tradi-tional liquid fuels. Energetic nanomaterials offer high reactivity,fast ignition, and fast energy release [5]. When mixed with liquidfuels, they can increase the volumetric energy density of the fuel,which is the most important parameter of fuel for high-speed pro-pulsion systems. The nanofluid-type fuels comprise a new class offuels and have been rarely studied. Jackson et al. [6] measured theignition delay time of slurries of n-dodecane and nano-aluminum

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m Ave., West Lafayette, IN494 0307.

nano-aluminum could substantially decrease the ignition delaytime. Using a simple hot-plate experiment, Tyagi et al. [7] foundthat the ignition probability for fuel mixtures that contained alu-minum nanoparticles was significantly higher than that of purediesel. Gan and Qiao [8] recently compared the burning character-istics of fuel droplets with nano- and micron-sized aluminum par-ticles. The results reveal that for the same solid loading rate andsurfactant concentration, the disruption and microexplosionbehavior of the micron suspension occurred later with much stron-ger intensity. Moreover, the authors show that particle collisionand aggregation behavior play an important role in the overallburning characteristics of fuel droplets.

In practical combustion systems such as diesel engines, gas tur-bines, and rocket engines, as well as oil-fired boilers, liquid fuel issprayed into the combustion chamber as a cloud of droplets [9].When the droplets are exposed to a hot environment, they evapo-rate quickly. In fact, droplet evaporation is an important process inthe combustion of liquid fuels and is also important for nanofluidswith applications in boiling heat transfer. However, studies on theevaporation behavior of nanofluids are rare. Chon et al. [10] inves-tigated the evaporation and dry-out characteristics of a stronglypinned water droplet with nanoparticles on a heated substrateand identified three periods: liquid dominant evaporation, dry-out progress, and nanoparticle strain. Sefiane and Bennacer [11]studied evaporation kinetics and wetting dynamics of ethanol-based sessile nanofluid droplets on rough heated substrates. Areduction of evaporation rate compared to the base fuel was foundduring the pinning phase. Considering three different types of

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Nomenclature

D droplet diameter (m)k Boltzmann constant (1.38 � 10�23 m2 kg s�2 K�1)K droplet evaporation rate (mm2/s)ni number of the i-fold aggregateri radius of the i-fold aggregate (m)Si breakup rate of the i-fold aggregate (s�1)t time (s)T temperature (K)

Greek symbolsaij collision efficiencybij collision frequency (m3/s)

Ui distribution functionl viscosity (kg m�1 s�1)

Subscriptsi i-fold aggregatej j-fold aggregate

4914 Y. Gan, L. Qiao / International Journal of Heat and Mass Transfer 54 (2011) 4913–4922

nanoparticles, Chen et al. [12] studied the effects of nanoparticleson the evaporation process of deionized water. They observed aunique behavior; that is, the evaporation rate changed from oneconstant to another after reaching a critical time, which they calltransition phenomena. This phenomenon was explained based onthe nanofluid’s apparent heat of vaporization.

Previous studies on the evaporation of nanofluids have providedimportant insights, but many questions remain. For example, themechanism responsible for the transition behavior Chen et al.[12] observed has not been well understood. Furthermore, the ef-fects of particle aggregation on droplet evaporation rate have notbeen carefully considered. We have shown in a previous study thatparticle aggregation has a significant impact on droplet combus-tion behavior [8]. Motivated by this, we investigated the evapora-tion characteristics of fuel droplets with the addition of energeticnanoparticles under various conditions.

The objectives of the paper are first to experimentally deter-mine the evaporation rate by considering various base fuels withaddition of nanoparticles at varying concentrations under naturaland forced convections; second, to model the dynamic particleaggregation process inside a vaporizing droplet and to determineits effect on the evaporation rate. The paper starts with a briefdescription of fuel preparation and characterization methods, afterwhich the experimental method is introduced. We then discuss theevaporation characteristics of different types of fuel droplets, e.g.,the effects of base fuel, particle concentration, and flow tempera-ture. In particular, the deviation from the D2-law phenomenon isreported and discussed. Lastly, we discuss a model that we builtbased on the population balance equation (PBE), which we solvedto simulate particle aggregation dynamics inside an evaporatingdroplet. The simulation results help to explain the D2-law devia-tion phenomenon observed in the experiment.

2. Experimental methods

2.1. Fuel preparation and characterization

Two liquid fuels, n-decane and ethanol, were considered as thebase fuel. Their physical properties are shown in Table 1. Alumi-num nanoparticles purchased from Nanostructured & Amorphous

Table 1Physical properties of n-decane, ethanol and Sorbitan Oleate.

Chemical formula Molecular weight

n-Decane C10H22 142Ethanol C2H6O 46Sorbitan Oleate C24H44O6 428

Materials, Inc. have a mean diameter of 80 nm. Fig. 1 shows anSEM photograph of the nano-Al samples. Most particles are spher-ical, and the size distribution is in the range of 35–100 nm.

The method used to disperse particles evenly in base fuel wasdescribed in a previous study [8]. Here we will only briefly discussthe preparation process. The challenge was to form a stable,homogenous mixture with a minimum degree of particle agglom-eration. Sonication [13] and addition of a surfactant are generallyused as a means to reduce particle agglomeration and to promotestabilization of the suspension. At first, particles were mixed withliquid fuels and vigorously stirred by hand. The mixture was thensonicated using an ultrasonicator; this was performed in an icebath for about 5 min. In some mixtures, a surfactant (Sorbitan Ole-ate) was added before sonication to promote chemical stabilizationof the suspension.

After sonication, suspensions of ethanol/nano-Al (1 wt.%) canlast for 24 h without obvious sedimentation. However, n-decane/nano-Al (1 wt.%) typically can remain stable for only 20 min, be-yond which sedimentation was observed. To promote suspensionstability, Sorbitan Oleate was added to them as a chemical agent.The maximum concentration of added surfactant was 0.5 wt.%,and with this the n-decane/nano-Al suspension can maintain sta-ble for more than 5 h, much longer than suspensions without sur-factant. This is because of the steric stabilization mechanism ofsurfactant [14]. We have explained in a previous study [8] that eth-anol-based suspensions can last much longer than n-decane-basedsuspension because ethanol has a higher viscosity than n-decaneand also because of ethanol’s ability to form gel structures aroundthe particles [15].

2.2. Experimental setup and data analysis

Fig. 2 shows a schematic of the droplet evaporation experiment.The setup consists of a droplet suspension system, a heated nitro-gen flow, a high-speed camera, and a data acquisition system.Droplets with a diameter of about 1 mm were suspended on thewelding point of a thin thermocouple (75 lm). To control the sizeof a suspended droplet, a syringe pump was adjusted to run at alow flow rate, which produced a droplet at a desired size at thetip of the needle. It was then transferred on the thermocouple. A

Boiling point (K at 1 atm) Viscosity (mPa s at 20 �C)

447 0.92352 1.2852 1200–2000

Fig. 1. SEM photograph of aluminum nanoparticles with a mean diameter of80 nm [8].

Y. Gan, L. Qiao / International Journal of Heat and Mass Transfer 54 (2011) 4913–4922 4915

glass shield was placed around the droplet to avoid disturbancesfrom the environment and to provide optical access.

Droplets were either vaporized under natural convection atroom temperature or under weak forced convection at elevatedtemperatures of up to 500 K. This was achieved by flowing a pre-heated nitrogen flow around the droplet. Nitrogen was heatedwhen flowing through a coil made of stainless steel that waswrapped with heating tape. The temperature of the heating systemwas controlled by a feedback temperature controller. Nitrogen wasselected because it can avoid potential oxidation and burning offuel droplets at higher temperatures. The heated nitrogen flowpassed a honeycomb and glass beads, which created a uniformlaminar flow around the droplet.

A high-speed digital camera coupled with a microlens was usedto record droplet evaporation history. An adjustable extension tubewas placed between the camera and the microlens to get a magni-fied view of the droplet. A light source was on the opposite side ofthe camera to backlight the droplet. A thin thermocouple was used

Fig. 2. Schematic of the

to suspend the droplet and also to record its temperature. One wasalso placed below the droplet to monitor the temperature of thenitrogen flow. Both were type-K thermocouples. The temperaturedata were acquired by a 1000-Hz data acquisition system synchro-nized with the high-speed camera.

After the droplet was placed on the tip of the thermocouple, thecamera and the temperature data acquisition system were simulta-neously triggered. For each experiment, about 6000 images andtemperature values were recorded to achieve a satisfactory timeresolution. The microlens and the camera were placed 10 cm fromthe droplet, and the magnification was kept constant. The resolu-tion of the high-speed camera was set to be 256 � 256 pixels with8 bits for each pixel. A grid with known size was placed in the sameposition as the droplet, and the spatial resolution of imaging wasthus determined to be 6–7 lm per pixel.

To postprocess the many images, a Matlab code was developedto determine droplet size history, from which droplet evaporationrate could be derived. A threshold value for pixel gray level wascarefully set to count the pixels in the droplet zone. The dropletsize was then calculated from the number of pixels. Lastly, at leastthree tests were carried out for each experimental condition to val-idate the repeatability of the experiment. The uncertainties of theevaporation rate measurements were estimated within ±3%.

3. Results and discussion

3.1. Typical droplet evaporation process

In this section, we will discuss the general characteristics of theentire droplet evaporation process, e.g., droplet size and tempera-ture histories, and how we used and interpreted the data. After thedroplet was positioned on the joining point of the thermocouples,data acquisition systems were triggered, and nitrogen was flowedthrough the test section.

Fig. 3 shows the typical droplet size and temperature historiesfor an ethanol-based fuel droplet under weak convection with anitrogen flow temperature of 335 K. The droplet contains 2.5%(by weight) aluminum nanoparticles with an average size of80 nm. Three distinct stages were identified based on the synchro-nized droplet size and temperature curves: natural convection,weak forced convection, and dry-out stage. The ethanol dropletwas initially under natural convection before the heated nitrogen

experimental setup.

Fig. 3. Typical droplet size and temperature history for ethanol-based fuel (2.5%nano-Al).


stream flowed through it. In this stage, the droplet size decreasesslightly as a result of natural evaporation. During the second stage,the heated nitrogen stream was applied, which resulted in a steeprise in the flow temperature. The droplet temperature was alsoslightly elevated. A steady-state evaporation process was estab-lished after a short time (about 3 s), in which the square of thedroplet size regresses linearly with respect to time. During thesteady-state evaporation stage, the flow temperature remained al-most the same, and the droplet temperature gradually increased by5 K. The overall temperature difference between gas flow anddroplet was about 40 K during this stage. In the final stage, all li-quid in the droplet had been vaporized, and an agglomerate ofabout 100 lm was left on the fiber. The flow and droplet temper-atures quickly dropped after the nitrogen flow was shut off.

Fig. 4 shows the typical droplet size and temperature historiesof an n-decane-based droplet under weak convection with a nitro-gen flow temperature of around 335 K. The mass ratio of the addednanoparticles is the same as in the ethanol-based droplet, 2.5%.Similarities are found between Figs. 3 and 4. However, differencesexist and are discussed below. First, the temperature differencebetween droplet and gas flow is about 5 K, lower than the 40 Kdifference for the ethanol-based droplet. Second, the droplet tem-perature rises to reach the same temperature of the gas flow before

Fig. 4. Typical droplet size and temperature history for n-decane-based fuel (2.5%nano-Al).

the dry-out stage. The droplet size regresses nonlinearly during thelatter stage of the evaporation process. Lastly, the lifetime of thedroplet is longer for the n-decane-based droplet than for the etha-nol-based droplet (210 s vs. 70 s) because of n-decane’s higherboiling point than ethanol’s. Having discussed the entire dropletevaporation process, it is important to note that for all experimentsonly the steady-state evaporation process was considered in theanalysis.

3.2. Evaporation of pure liquid droplets

The evaporation rate of pure liquid was measured first as abaseline to understand the effect of various added nanoparticleson droplet evaporation behavior. Fig. 5 shows the evaporation rateof an ethanol droplet under natural convection at 300 K and underweak forced convection at various flow temperatures (300–380 K).It is clear that the droplet size variation with time follows the clas-sical D2-law under all conditions, which comes from Langmuir’soriginal work [16]. Note that the droplet diameter D was scaledby the original droplet diameter D0, as suggested by the classicaltheory of droplet evaporation [9].

3.3. Evaporation of ethanol-based droplets with the addition ofnanoparticles

In this section, we will discuss the evaporation characteristics ofethanol droplets containing Al nanoparticles at various concentra-tions under natural and forced convections. Fig. 6 shows the dropletsize as a function of time for ethanol droplets with the addition of0.5 wt.% nano-Al under various temperatures. Under weak forcedconvection at temperatures of 330 K and 380 K, the droplet size re-gresses following the classical D2-law. The evaporation rate underweak forced convection at room temperature (300 K) also approxi-mately follows the D2-law. The variation of droplet sizes under nat-ural convection at room temperature (300 K), however, shows afeature that deviates from the D2-law – the curve is ‘‘bent.’’ Theinstantaneous droplet evaporation rate was plotted here as a func-tion of time in Fig. 7 (dash line). It can be seen that this ratedecreases all the way until the very end where it shows a slightbounce back. The minimum evaporation rate occurs aroundt = 250 s, and a 36% reduction was observed (to 0.0016 mm2/s, from0.0025 mm2/s).

The results for ethanol droplets containing a higher concentra-tion (2.5 wt.%) of nano-Al particles are shown in Fig. 8. It seems

Fig. 5. Evaporation of ethanol droplets under different temperatures.

Fig. 6. Evaporation of ethanol droplets with 0.5% nano-Al under differenttemperatures.

Fig. 7. Evaporation rate of ethanol droplets with nano-Al under natural evaporationat 300 K.

Fig. 8. Evaporation of ethanol droplets with 2.5% nano-Al under differenttemperatures.


that the droplet-size history follows the D2-law under weak forcedconvection of various flow temperatures (300 K, 330 K, and 380 K).Under natural convection at room temperature (300 K), however,the square of the droplet size clearly deviates from the D2-law.Also, the bending of the curve is more prevalent than for the drop-let containing 0.5 wt.% nano-Al particles shown in Fig. 6. In otherwords, the deviation from the D2-law is more significant at higherparticle concentrations. The instantaneous evaporation rate for thedroplet with 2.5 wt.% nano-Al was also plotted as a function oftime in Fig. 7 (solid line). The evaporation rate decreases continu-ously. A reduction of about 50% in the evaporation rate was ob-served (to below 0.0015 mm2/s, from 0.0030 mm2/s), which ishigher than the reduction for the droplet with 0.5 wt.% nano-Al.

3.4. Evaporation of n-decane-based droplets with the addition ofnanoparticles

In this section, we will discuss the evaporation characteristics ofn-decane-based droplets containing nanoparticles. Compared toethanol, n-decane is a high-boiling-point fuel and requires a longertime for complete evaporation. Fig. 9 shows the droplet size histo-ries of n-decane-based fuel droplets with the addition of 0.5 wt.%nano-Al particles under various conditions. At higher temperatures(320 K and 380 K) under weak forced convection, the droplet sizevariation obeys the D2-law. However, the law’s deviation phenom-enon occurs under weak convection at room temperature (300 K).As a comparison, under the same condition the evaporation of eth-anol-based droplets containing either 0.5 wt.% or 2.5 wt.% nano-Alparticles still follows the D2-law. Note that the evaporation processof n-decane-based fuels under natural convection at 300 K is notshown in the figure because the droplet lifetime is much longer(about 2 h) than the others, but the D2-law deviation phenomenonis quite obvious under this condition.

The results for n-decane-based droplets containing 2.5 wt.%nano-Al particles are shown in Fig. 10. The D2-law deviation phe-nomenon occurs at 300 K and 320 K under weak forced convection.Moreover, the curve at 300 K is more ‘‘bent’’ for 2.5 wt.% nano-Alparticle addition than from the 0.5 wt.% nano-Al particle addition.This is consistent with the results of the ethanol-based droplets –the deviation from the D2-law is more significant at higher particleconcentrations. Under even higher temperatures (such as 380 K),the droplet size history follows the D2-law. The droplet lifetimeof n-decane-based fuels is found to be much longer than that ofethanol-based fuels (650 s vs. 100 s).

Fig. 9. Evaporation of n-decane droplets with 0.5% nano-Al under differenttemperatures.

Fig. 10. Evaporation of n-decane droplets with 2.5% nano-Al under differenttemperatures.


Similarly, the instantaneous droplet evaporation rate for n-dec-ane-based droplets with 0.5 wt.% and 2.5 wt.% nano-Al additionwas plotted as a function of time in Fig. 11. For both, the evapora-tion rate decreases with time. For the n-decane-based droplet with0.5 wt.% nano-Al, a 48% reduction was observed (to 0.00073 mm2/s, from 0.0014 mm2/s). For the n-decane-based droplet with2.5 wt.% nano-Al, a higher (53%) reduction in evaporation ratewas observed (to 0.0008 mm2/s, from 0.0017 mm2/s).

3.5. Comparison with previous experiments

In summary, the experimental results show that the deviationfrom the D2-law phenomenon is more significant at higher particleconcentrations than at lower ones, assuming that the other condi-tions (such as base fluid and flow convection environment) are thesame. The deviation is more obvious at lower convection temper-atures than at higher ones for the same base fluid and at the sameparticle concentration. With regard to various base fluids, the devi-ation is more obvious for higher boiling point fuels than for lowerones. Although several factors may contribute to the D2-law devi-ation behavior, there is a common feature for all the experimentsin which deviation was observed – the droplet lifetime is much

Fig. 11. Evaporation rate of n-decane droplets with nano-Al under weak evapora-tion at 300 K.

longer, compared to the droplet lifetime in the experiment forwhich the D2-law obeys. The droplet lifetime seems to play a rolein whether the evaporation follows or deviates from the D2-law.

Chen et al. [12] studied the effect of three nanoparticles (lapo-nite, Fe2O3, and Ag) on the evaporation rate of deionized water un-der natural convection at room temperature. The results show thatthe evaporation may be enhanced or slowed down, depending onthe type of nanoparticle and whether it is used in combinationwith the surfactant PVP. In particular, the evaporation rate of var-ious Ag and Fe2O3 nanofluids goes through a transition from oneconstant value to another during the evaporation process. Theauthors explained the effect of various nanoparticles on evapora-tion from the perspective of apparent heat of evaporation. Theaddition of nanoparticles generally increases the apparent heat ofevaporation and thus reduces the evaporation rate. The authorsalso suggested that the transition phenomenon was caused bythe increasing particle concentration during the droplet evapora-tion process as the droplet volume became smaller.

The present experiment deals with different base fluids (mainlyliquid hydrocarbon fuels) and nanoparticles (reactive metal parti-cles) with an interest in burning these nanofluid-type fuels. Further,it considered both natural convection and weak forced convection.Some of our results are consistent with the observations of Chenet al. [12]. For instance, nanofluid droplets vaporize as a pure liquiddroplet at the beginning stage, and nanoparticles inside the dropletgenerally tend to slow down the droplet evaporation rate. However,we did not observe the transition phenomenon, that is, the evapo-ration rate K changes from one constant value to another constantafter a critical time is reached. Instead, our finding was that theevaporation rate K varies throughout the entire evaporation processin situations in which the D2-law does not apply.

Possibly the addition of particles changes the apparent heat ofevaporation and thus modifies the evaporation behavior. Particlescould also change the viscosity and surface tension of the fluid,which in turn could impact evaporation. Nevertheless, the effectof particle aggregation behavior on droplet evaporation has notbeen seriously considered. During the evaporation process of amulticomponent (liquid + solid) droplet, there are two majorevents: fluid dynamics and particle dynamics. Fluid dynamicsmeans the motions associated with the fluid component, includingdroplet surface regression, internal circulation, and liquid diffusionfrom inside to surface. Particle dynamics means the motions asso-ciated with particles, e.g., particle collision, particle aggregation,and aggregate breakup. The motions of the two components are in-deed coupled. For instance, fluid motion could impact particleaggregation. On the other hand, particle aggregation could also af-fect fluid dynamics and thus impact evaporation.

Motivated by this, we modeled the particle aggregation processinside a droplet. Our goal here is not to develop or validate aggre-gation kinetics. In fact, simulating the detailed transient aggrega-tion process is a challenging task. This is because the rateparameters and coefficients that appeared in the PBE are largelyunknown, e.g., the collision efficiency between clusters with vari-ous sizes and structures, and the breakup coefficient of largerstructures to smaller ones. This is mainly responsible for theremarkable differences between theoretical predictions and exper-imental results. Another factor making the simulations extremelychallenging is that primary particles can form aggregates that havevarious shapes and densities. And it is impossible to predict the de-tailed structure of the aggregates. To solve this problem, aggre-gates are usually modeled as fractals using fractal dimension asan additional variable in the population balance equation [17].Our goal here is to qualitatively understand the aggregation pro-cess inside an evaporating droplet and how this process couldpotentially impact the droplet evaporation rate, especially theD2-law deviation behavior.


3.6. Modeling of the particle aggregation process

The goal of the modeling work is to understand the transientaggregation process of nanoparticles inside a vaporizing dropletunder natural or forced convections. This may help to explain theexperimental observations, especially the D2-law deviation phe-nomenon. In the present study, the evolution history of particleaggregates (including aggregation size and distribution) were ob-tained by solving the PBE. This is a classical method of modelingthe evolution of particulate systems by accounting for variousevents taking place between particles in the population, e.g., aggre-gation and breakage. Kumar and Ramkrishna [18] has provided acomprehensive review of the application of population balanceequations in particulate systems. In the following, we will describeformation of the model and the numerical method used to solvethe equation sets, and then we will discuss the results and howthey can be used to understand the effect of particle aggregationon droplet evaporation rate.

3.6.1. Description of the model and the numerical methodParticle aggregation is a result of collisions between particles.

Thus the aggregation process depends on collision frequency andcollision efficiency. According to Fendler and Dekany [19], thereare three major transport mechanisms responsible for particleaggregation in nanofluids: Brownian diffusion, fluid motion, anddifferential setting. The Brownian motion means the randommovements of small particles. It is also called perikinetic collisionsand has been studied widely by the nanoscience community. Thesecond mechanism (fluid motion) means the relative motion ofparticles induced by fluid transport, e.g., shear flow and agitation.This mechanism is also called orthokinetic aggregation [19], andit can significantly increase the chance of collisions between parti-cles, especially for large particles. The third mechanism is becauseof gravity – when particles of different sizes and densities are set-tling in a nanofluid, they tend to capture other particles as they fall.

The model describes the dynamic aggregation process of Alnanoparticles (80 nm) in a vaporizing fuel droplet (ethanol andn-decane), as shown in Fig. 12. To match the experimental condi-tions, the droplet in the modeling has sphere geometry with adiameter of 1 mm. Initially, Al nanoparticles were uniformly dis-tributed within the droplet. The collision frequency between parti-cles becomes higher as the droplet is being vaporized becauseparticle concentration becomes higher. After a sufficiently long

Fig. 12. Particle aggregation model in a vaporizing droplet.

time, it is possible that all particles adhere and form one or a fewlarge agglomerates when all liquid is consumed.

According to Smoluchowski’s classical work [20], the PBE,which describes the birth and death of aggregate as a result of col-lision and breakup, can be written as

dni

dt¼ 1

2

Xi�1

j¼1

aj;i�jbj;i�jnjni�j �Xn

j¼1

aijbijninj � Sini þXn

j¼1

CijSjnj ð1Þ

The left side of the equation expresses the variation rate of thenumber of i-fold aggregates, where ni is the number of the i-foldaggregate and t is time. Note that the i-fold aggregate means a clus-ter that contains i original particles. The mass of the cluster is itimes the mass of the original single particle. In the beginning, par-ticles of the same size are uniformly distributed within the droplet.Throughout the evaporation process, aggregates of various sizeswill be formed. For the aggregation of solid particles, no coalescencecan occur, and thus the resulting clusters may have different struc-tures [19]. It is extremely difficult to provide a detailed descriptionof the aggregate structure, especially when hundreds or thousandsof particles are involved. Thus we defined the size of the i-foldaggregates using a simplified way, as follows:

43pr3

i ¼ i43pr3

1 ð2Þ

where r1 is the initial particle radius and ri is the nominal radius ofthe i-fold aggregates.

The first term on the right of the equation represents the rate offormation of the i-fold aggregates as a result of collision betweenany pair of aggregates, i–j and j. The collision efficiency and colli-sion frequency between i and j folds are aij and bij. Summation inthis way actually counts each collision twice, thus a factor of 0.5is applied. The second term on the right of the equation accountsfor loss of the i-fold aggregates as a result of collision and aggrega-tion with other aggregates. The third term describes the breakup ofthe i-fold, where Si represents the breakup rate. The fourth termstands for the contribution to i-fold from the breakup of any otherfolds, where Uij is the distribution function of aggregate fragments.

Several assumptions were made in the model. First, only two-body collisions were considered following Smoluchowski’s classicwork [20]. This is a reasonable assumption because for dilute sus-pensions, the collisions by three or more bodies have a much lowerfrequency than the two-body collisions, and thus they can be ne-glected. Also, the particle aggregation process was simulated onlyfor the first 50% of the droplet steady-state evaporation process.At a later stage, the particle concentration can be much higher be-cause the droplet volume is decreased; thus the two-body collisionassumption may not be valid under these conditions.

Second, we have shown in a previous study [8] that for nano-suspensions, the Brownian motion is mainly responsible for parti-cle collision and aggregation. The rates of particle collision as aresult of fluid transport (such as droplet surface regression andinternal circulation) and differential settling are much smaller.However, for suspensions with micron-sized or larger particles,the fluid transport is more important than the Brownian motion.Because nanosized particles are of interest and were used in theexperiments, only aggregation kernels resulting from the Brownianmotion were considered in the model. Fluid motion can give a sig-nificant increase in the rate of particle intercollision where the col-loids were subject to shear, e.g., by vigorous stirring or byturbulence [19]. In the present paper, however, droplets were un-der natural or very weak convection. The internal circulationcaused by relative velocity between the gas and liquid phases isweak compared to that under high Reynolds number convections[21]. Thus fluid motion inside the droplet is less likely to make asignificant contribution to particle collisions. It is important to note

Fig. 13. Time history of number for i-fold aggregates inside an ethanol-baseddroplet with 0.5 wt.% nano-Al addition under natural convection of 300 K. At t = 0 s,the droplet has a diameter of 1 mm, the particle size is 80 nm, and there are 3billion particles present within the droplet.


that when the size of the aggregates is sufficiently large, the orth-okinetic aggregation because of fluid transport can become domi-nant over perikinetic collisions.

Lastly, we assume that particle aggregation be a one-way pro-cess without breakup. This means that the last two terms on theright side of PDE are zero. This is a reasonable assumption becausethe breakup of aggregates is mainly brought about by hydrody-namic stresses [22], which is the fluid shear induced by fluid mo-tion. The droplets in this study were vaporized under natural orvery weak convection. Thus fluid shear induced by internal motionis weak, especially compared to that under high Reynolds numberconvection. In all, the breakup of particle aggregates is less likely tomake a significant contribution to particle collisions.

The collision frequency bij between the i and j folds can be ex-pressed as [20]

bij ¼2kT3lðri þ rjÞ2

rirjð3Þ

where k is the Boltzmann constant, T is the temperature, l is the vis-cosity of the liquid, and ri and rj are the radii of particles i and j,respectively. The collision efficiency aij is assumed to be unity, whichmeans that every collision results in an attachment of particles. Inreality, collision efficiency can be significantly reduced because ofrepulsive forces from colloidal interaction, e.g., the steric interactionas a result of adding a surfactant. Another main factor is the hydro-dynamic effect. As particles become very close to one another, it be-comes increasingly difficult for the base liquid between them todrain out. This will hinder the approach of the particles and thus re-duce collision efficiency. Overall, collision efficiency mainly dependson the forces between the particles and the hydrodynamic effect. Itis extremely difficult to provide an accurate expression of collisionefficiency [19]. In a recent study by Chen and You [23], differentforces between the particles were considered to determine it,including van der Waals force, elastic deformation force, and electro-static force. The results show that the collision efficiency increaseswhen the particle size decreases. When the size of the particle is lessthan 100 nm, the collision efficiency is approaching unity. Here weused unity for the collision efficiency as an ideal situation. Thismeans that the real aggregation process could be much slower thanthe ideal process we are simulating here.

Based on these assumptions, the discrete form of the populationbalance equations for every i-fold was numerically solved to getthe temporal evolution of the aggregates with various size. Theordinary differential equations were solved numerically usingthird-order Runge–Kutta methods. The initial values (for example,with 0.5 wt.% nano-Al uniformly distributed within a droplet withan initial diameter of 1 mm) were set as n1 = 3 � 109, n2 = n3 =� � � = n3�109 = 0 at t = 0, where ni means the number of i-fold in thedroplet. It is noted that droplet size regression was also consideredin the aggregation model, and this will be discussed further in thenext section.

3.6.2. Modeling resultsWe first discuss the aggregation process inside an ethanol-based

droplet. Fig. 13 shows the time history of the aggregate number for7 representative aggregates ranging from 1-fold to 10,000-fold. Theinitial condition is the same as that in Fig. 6 (line with square dot)�an ethanol-based droplet with 0.5 wt.% nano-Al addition under nat-ural evaporation at 300 K. Under such circumstances, 3 billion par-ticles were suspended inside the droplet initially at t = 0 s. The term‘‘i-fold’’ means the aggregates containing i particles.

The 1-fold particles (primary particles) disappear quickly byforming two- or more-particle clusters, as indicated by an earliersteep drop of the 1-fold curve (t < 5 s). The decrease rate of the 1-fold number slows at later times (t > 20 s). By the end of the compu-

tation time, which is about 60 s, only 147 individual particles areleft. All other folds have the same trend: the number of the i-fold in-creases first, reaches a maximum value at a critical time, and thendecreases gradually. The critical time occurs later for the higher foldthan it does for the lower fold. For example, the critical time corre-sponding to the maximum number occurs at t = 1.8 s for 100-fold,at t = 8.95 s for 1000-fold, and at t = 83 s for 8000-fold. As time goeson, the population of smaller clusters decreases and that of largerclusters increases as a result of collisions between clusters. Anotherobservation is that the number of all folds tends to reach an equilib-rium point around 100 s, shown as a ‘‘band shape’’ in Fig. 13.

The particle aggregation process inside an evaporating dropletunder natural convection is seen to be a continuous and slow pro-cess. Larger aggregates were produced, and smaller aggregates dis-appeared because of collisions among them. The structures of largeaggregates can be complicated, resulting from the random combi-nation of individual particles that may interlink with one anotherto form even more-complicated structures. The existence of largeaggregate structures tends to inhibit internal diffusion of the liquidfuel from inside toward the droplet surface. Consequently, the drop-let evaporation rate is reduced. The reduction in evaporation rate ismore significant when larger aggregates are formed. This explainsthe D2-law deviation phenomenon observed in the experiment.

Next, we discuss the effect of particle concentration on dropletevaporation. Fig. 14 shows the aggregate number as a function oftime for 7 representative aggregates ranging from 1-fold to10,000-fold for an ethanol-based droplet with 2.5 wt.% nano-Aladdition under natural evaporation at 300 K. Fig. 14 shares somesimilarities with Fig. 13 (with 0.5% addition). However, there aredifferences. First, the time needed for the i-fold aggregate to formis shorter at higher particle concentrations than at lower ones. Forexample, the 5000-fold aggregates start to form at t = 4.21 s for the2.5 wt.% nano-Al addition case, and at t = 18.05 s for the 0.5 wt.%nano-Al addition case. Moreover, the critical time correspondingto the maximum number of each fold appears earlier if the concen-tration of particles is higher. When the particle concentration ishigher, the collision frequency between particles is also higher.As a result, large aggregate structures will form at earlier timesand have a higher number density. This means that the suppres-sion of droplet evaporation because of these large structures willoccur earlier and will be more serious. And this explains the exper-imental observation that the evaporation curve under naturalevaporation in Fig. 8 shows a more-obvious departure from theD2-law compared to that in Fig. 6.

Fig. 14. Time history of number for i-fold aggregates inside an ethanol-baseddroplet with 2.5 wt.% nano-Al addition under natural convection of 300 K. At t = 0 s,the droplet has a diameter of 1 mm, the particle size is 80 nm, and there are 15billion particles present within the droplet.

Fig. 16. Time history of number for i-fold aggregates inside an n-decane-baseddroplet with 2.5 wt.% nano-Al addition under weak forced convection of 300 K. Att = 0 s, the droplet has a diameter of 1 mm, the particle size is 80 nm, and there are 3billion particles present within the droplet.


Next we will discuss the effect of the base fuel on droplet evap-oration. Fig. 15 shows the temporal evolution of the aggregatenumber for 7 representative aggregates ranging from 1-fold to10,000-fold for an n-decane-based droplet with 0.5 wt.% nano-Aladdition under weak convection at 300 K. Note that this corre-sponds to the experimental result shown in Fig. 9 (line with squaredot). Fig. 16 shows the computational results for an n-decane-based droplet with 2.5 wt.% nano-Al, which corresponds to theexperimental results shown in Fig. 10 (line with square dot). Sim-ilar results can be concluded for the n-decane-based droplets inFigs. 15 and 16. For instance, at higher particle concentrationsthe departure from D2-law is more obvious and the reduction ofthe evaporation rate is bigger. Comparing to the results inFig. 13, we can see that large aggregates take a longer time to formfor n-decane-based droplets than for ethanol-based droplets. Forexample, the time needed to form the 10,000-fold aggregates is72.18 s for an n-decane-based droplet, which is 42.10 s for an eth-anol-based droplet. Note the lifetime of the n-decane-based drop-let (about 650 s) is longer than that of the ethanol-based droplet(about 350 s).

Fig. 15. Time history of number for i-fold aggregates inside an n-decane-baseddroplet with 0.5 wt.% nano-Al addition under weak forced convection of 300 K. Att = 0 s, the droplet has a diameter of 1 mm, the particle size is 80 nm, and there are 3billion particles present within the droplet.

In summary, the modeling results show that the particle aggre-gation process inside an evaporation droplet is a continuous andslow process under natural convection or weak forced convection.During earlier times, smaller-sized aggregates that consist of only afew particles dominate the population. Later, larger-sized aggre-gates start to form, and they may eventually dominate the popula-tion. The present experiment deals with the evaporation ofnanofluid-type fuels at natural and weak convections. The charac-teristic time required to form large aggregates is much longer thanunder strong convection, e.g., combustion. Previous studies ondroplet combustion of nanofluid fuels have shown that a shellwas formed during the combustion process as a result of particleaggregation, and the existence of the shell inhibited the diffusionprocess of the base fuel and thus reduced the burning rate [8].We can expect that droplet evaporation will be affected when largeaggregate structures are present within the droplet. The largeaggregates inhibit diffusion and thus decrease the evaporationrate. If the droplet lifetime is longer than or at least comparableto the characteristic aggregation time, the particle aggregation pro-cess will have an effect on the evaporation. Moreover, the contin-uous change in droplet evaporation rate would be expected as aresult of the continuous aggregation process: the number of largeraggregate structures increases with time; the number of smalleraggregates decreases. If the droplet lifetime is shorter than thecharacteristic aggregation time, it means that during the dropletevaporation process, smaller-sized aggregates dominate the popu-lation, and larger ones have not yet had sufficient time to form. Un-der these circumstances, particle aggregation would have the leastimpact on droplet evaporation because the aggregate structurethat consists of only a few particles would have the least impacton diffusion. This explains why under certain conditions (such asstrong forced convection at higher temperatures, or with the useof lower-boiling-point fuels as the base fluid), deviation from theD2-law was not observed because the droplet evaporates fasterthan the particle aggregate in such conditions.

4. Conclusions

The effect of added aluminum nanoparticles on the evaporationcharacteristics of fuel droplets were investigated under both natu-ral and forced convections. The droplet evaporation rate was mea-sured by considering various base fuels, at varying particleconcentrations and at convective flow temperatures. Furthermore,


the particle aggregation process was numerically modeled tounderstand the potential role of particle aggregation on dropletevaporation rate. The evolution history of particle aggregates(including aggregation size and distribution) was obtained by solv-ing the PBEs, which accounts for various events taking place be-tween particles in the population, e.g., aggregation and breakage.The major conclusions are as follows.

As expected, the evaporation of pure liquid droplets such as eth-anol and n-decane follows the classical D2-law under all test con-ditions. For ethanol-based nanofluids under weak convection atthree flow temperatures (300 K, 330 K, and 380 K), the evaporationalso follows the D2-law. A departure from it was observed undernatural convection at 300 K. The droplet evaporation rate wasgradually slowed. The reduction in the evaporation rate is 30%for 0.5 wt.% particle addition and 50% for 2.5 wt.% particle addition.

For n-decane-based nanofluids, the D2-law holds when dropletsare vaporized under weak convection at higher temperatures (e.g.,380 K). A departure from the D2-law was observed under both nat-ural convection at 300 K and weak forced convection at 300 K forn-decane droplets with 0.5 wt.% nano-Al. For n-decane dropletswith 2.5 wt.% nano-Al, a departure was observed under weak con-vection at 300 K and 320 K. These results show that the D2-lawdeviation phenomenon is more prevalent at higher particle con-centrations and for base fluids that have higher boiling points. Acommon feature among the conditions for which the deviationwas observed is the long droplet lifetime.

The modeling results suggest that the D2-law deviation phe-nomena and the continuous decrease in droplet evaporation ratecould be a result of particle aggregation inside a vaporizing droplet,which is a rather slow process under natural or weak convection. Ifthe droplet lifetime is longer or comparable to the characteristicaggregation time, large aggregates are formed during the dropletevaporation process, which could inhibit diffusion and thus reducethe evaporation rate. On the contrary, if the droplet lifetime isshorter than the characteristic aggregation time, smaller-sizedaggregates dominate the population, and they have the least im-pact on diffusion and evaporation. In such circumstances, largeraggregates that suppress droplet evaporation have not yet had suf-ficient time to form; deviation from the D2-law was therefore notobserved in these circumstances.

Acknowledgment

This work has been supported by the Army Research Office(ARO) with Dr. Ralph Anthenien as the technical monitor.

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